Algebraic classification of abelian varieties over finite fields

• Applicant: Dr. Stefano Marseglia
• Subject Area: Mathematics
• Term: since 2026
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 584021622

Some of the most extensively studied objects at the intersection of number theory and algebraic geometry are abelian varieties, which are projective varieties whose points form a group. Abelian varieties have a very rich algebraic structure which makes them fundamental tools for understanding the geometry and arithmetic of curves through their Jacobians. Moreover, abelian varieties of dimension 1, also known as elliptic curves, defined over finite fields are the basis for some of the most used cryptosystems. Because of these reasons, and many more, understanding the abelian varieties is important not just from the theoretical point of view, but also for their practical applications. Even if abelian varieties have been extensively studied, they still remain quite elusive. For example, in general, the known methods to describe them as subvarieties of a projective space output equations that, already in dimension 2, are so cumbersome that they are not practical to use. Our guiding goals are to concretely represent abelian varieties defined over a finite field, and classify them up-to-isomorphism, together with their polarizations. By achieving these objectives, we will fill some important gaps in our current understanding of many invariants attached to the abelian varieties, like the p-rank or the Newton polygon and the induced stratifications on the corresponding moduli spaces, which are still rather mysterious. Finally, improving our understanding of the isomorphism classes of abelian varieties has potential applications in post-quantum cryptography, via the so-called isogeny graphs, and in the theory of error-correcting codes, via the so-called AG codes. Our first step is to construct equivalences from the category of abelian varieties over a finite field to computable categories of modules. In the literature, one can find several equivalences analogous to the one we are after in this project, but they either are defined only over subcategories of the abelian varieties, or the target category consists of modules over a ring which is too complicated or too abstractly defined to be practical for effective computations. The second step of the project is to develop methods to classify the modules in the target category, theoretically in terms of numerical invariants, or algorithmically using a computer. In short, this project aims at building new bridges between the arithmetic geometry world of the abelian varieties over finite fields and the algebraic world of modules, where it is easier to make effective computations.

• DFG Programme: Research Grants